Problem 35 - Algebra 2

Question

Solve each equation. Check your solutions. \(\log _{2}(12 b-21)-\log _{2}\left(b^{2}-3\right)=2\)

Step-by-Step Solution

Answer

The solution is \( b = \frac{3}{2} \).

Step 1: Use Logarithm Properties

Apply the quotient property of logarithms: \( \log_a (x) - \log_a (y) = \log_a \left( \frac{x}{y} \right) \). Thus, the given equation \( \log _{2}(12b-21)-\log _{2}(b^2-3) = 2 \) can be rewritten as:\[ \log_{2} \left( \frac{12b-21}{b^2-3} \right) = 2. \]

Step 2: Convert to Exponential Form

Convert the logarithmic equation to an exponential form to eliminate the logarithm. If \( \log_b(x) = y \), this implies \( x = b^y \). Therefore:\[ \frac{12b - 21}{b^2 - 3} = 2^2 \]This simplifies to:\[ \frac{12b - 21}{b^2 - 3} = 4. \]

Step 3: Solve the Rational Equation

Multiply both sides by \( b^2 - 3 \) to eliminate the denominator:\[ 12b - 21 = 4(b^2 - 3). \]Expand the right side:\[ 12b - 21 = 4b^2 - 12. \]Rearrange the equation to form a standard quadratic equation:\[ 4b^2 - 12b + 9 = 0. \]

Step 4: Solve the Quadratic Equation

Factor or use the quadratic formula to solve \( 4b^2 - 12b + 9 = 0 \). The quadratic formula is: \( b = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \) with \( A = 4 \), \( B = -12 \), and \( C = 9 \).Calculate:\(B^2 - 4AC = (-12)^2 - 4\cdot4\cdot9 = 144 - 144 = 0.\)Since the discriminant is zero, there is one solution:\(b = \frac{-(-12)}{2\cdot4} = \frac{12}{8} = \frac{3}{2}.\)

Step 5: Check the Solution

Substitute \( b = \frac{3}{2} \) back into the original equation to verify:Calculate terms:\( 12b - 21 = 12\left(\frac{3}{2}\right) - 21 = 18 - 21 = -3\)\( b^2 - 3 = \left(\frac{3}{2}\right)^2 - 3 = \frac{9}{4} - 3 = \frac{9}{4} - \frac{12}{4} = -\frac{3}{4}\)If both terms are negative, calculate:\( \log_2 \left( \frac{-3}{-\frac{3}{4}} \right) = \log_2 \left( 4 \right) = 2,\)which matches the original equation. Hence, the solution is correct.

Key Concepts

Logarithm Properties

Logarithm properties are handy tools in simplifying complex logarithmic equations. One crucial property used in simplifying the problem is the **quotient property**, which states:

• \( \log_a (x) - \log_a (y) = \log_a \left( \frac{x}{y} \right) \)

This allows us to combine two logarithms into a single term, making it simpler to handle.
In the provided solution, the equation \( \log _{2}(12b-21)-\log _{2}(b^2-3) = 2 \) is rewritten using this property as \[ \log_{2} \left( \frac{12b-21}{b^2-3} \right) = 2. \]
This single logarithmic equation is more manageable and sets the stage for further simplification. Understanding and employing logarithm properties is pivotal in solving equations of this type.

Exponential Form

Transforming a logarithmic equation into an exponential form is a crucial step in existing mathematical problems. The transition from logarithmic to exponential form relies on the basic principle that if \( \log_b(x) = y \), then \( x = b^y \).
This switch effectively removes the logarithm, allowing for further algebraic manipulation.
In the given exercise, transitioning from \( \log_{2} \left( \frac{12b-21}{b^2-3} \right) = 2 \) leads us to:

• \( \frac{12b - 21}{b^2 - 3} = 2^2 \),
• which simplifies to \( \frac{12b - 21}{b^2 - 3} = 4 \).

By understanding exponential forms, students can seamlessly convert and simplify these equations, preparing them for solving rational equations.

Quadratic Equation

A quadratic equation is a polynomial equation of degree 2, typically in the form \( ax^2 + bx + c = 0 \). Solving quadratic equations is an essential skill in algebra, and there are several methods to solve them, including:

• Factoring,
• Using the quadratic formula,
• Completing the square.

In our exercise's context, the equation transforms into a quadratic form as:\[ 4b^2 - 12b + 9 = 0. \]To find its solution, we use the quadratic formula \( b = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \), where \( A = 4 \), \( B = -12 \), and \( C = 9 \).
Calculating the discriminant, \( B^2 - 4AC \), and finding it to be zero indicates one unique solution.
Maneuvering through quadratic equations allows solving for variables systematically and is essential for solving the rational equation.

Rational Equation Solving

Solving rational equations often involves finding an equation where variables appear in the numerator and the denominator. The process generally requires:

• Eliminating the fractions by multiplying through by a common denominator,
• Simplifying the equation,
• Solving the resulting equation.

After re-writing the logarithmic expression to its exponential form in the original exercise, we are left with the rational equation:\[ \frac{12b - 21}{b^2 - 3} = 4. \]Multiplied through by \( b^2 - 3 \), the equation simplifies to a quadratic form:\[ 12b - 21 = 4(b^2 - 3). \]Like all rational equations, it’s essential to verify solutions to ensure they don’t lead to undefined fractions. This step guarantees that the solution aligns with the constraints of the original equation.